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Convergence of the viscosity solutions for weakly strictly hyperbolic conservation laws*

Published online by Cambridge University Press:  14 November 2011

Zhu Changjiang
Zhu Changjiang
Affiliation:
Wuhan Institute of Mathematical Sciences, Chinese Academy of Sciences, P.O. Box 71007, Wuhan 430071, P.R., China
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Synopsis

This paper is an extension of papers [14–16]. Using the theory of compensated compactness, we establish the convergence of the uniformly bounded approximate solution sequence for a class of ‘weakly strictly hyperbolic’ conservation laws.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 1103 - 1110
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Ball, J. M.. Convexity conditions and existence theorems in the nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
2Chen, G. Q.. Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics, III. Acta Math. Sci. 6 (1986), 75–120.CrossRefGoogle Scholar
3Chuch, K. N., Conley, C. C. and Smoller, J. A.. Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26 (1977), 372–411.Google Scholar
4Dafermos, C. M.. Estimates for conservation laws with little viscosity. SIAM J. Math. Anal. 18 (1987), 409–21.CrossRefGoogle Scholar
5Ding, X. X., Chen, G. Q. and Luo, P. Z.. Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics, I, II. Acta Math. Sci. 5 (1985),483–500; 501–40.Google Scholar
6Diperna, R. J.. Convergence of approximate solutions to conservation laws. Arch.Rational Mech. Anal. 82 (1983), 27–70.CrossRefGoogle Scholar
7Diperna, R. J.. Convergence of the viscosity method for isentropic gas dynamics. Comm. Math. Phys. 91 (1983), 1–30.CrossRefGoogle Scholar
8Frid, H. and Santos, M. M.. The Cauchy problem for the systems . J. Differential Equations 111 (1994), 340–59.CrossRefGoogle Scholar
9Kan, P. T.. On the Cauchy problem of a 2 × 2 system of nonstrictly hyperbolic conservation laws (Ph.D. Thesis, Courant Institute, New York University, 1989).Google Scholar
10Klainerman, S. and Majda, A.. Singular perturbations of quasilinear hyperbolicsystems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34 (1981), 481–524.CrossRefGoogle Scholar
11Lax, P. D.. Shock waves and entropy. In Contributions to Nonlinear Functional Analysis, 603–34 (New York: Academic Press, 1971).CrossRefGoogle Scholar
12Lin, P. X.. Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics. Trans. Amer. Math. Soc. 329 (1992), 377–413.CrossRefGoogle Scholar
13Liu, T. P.. Solutions in the large for the equations of non-isentropic gas dynamics. Indiana Univ. Math. J. 26 (1977), 147–77.CrossRefGoogle Scholar
14Lu, Y. G.. Convergence of the viscosity method for a nonstrictly hyperbolic conservationlaw. Comm. Math. Phys. 150 (1992), 59–64.CrossRefGoogle Scholar
15Lu, Y. G.. Convergence of the viscosity method for some nonlinear hyperbolic systems. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 341–52.CrossRefGoogle Scholar
16Lu, Y. G., Z. Wang, Zhu, C. J. and Zhao, H. J.. Convergence of the viscosity solutions for a nonstrictly hyperbolic system (to appear).Google Scholar
17Murat, F.. L'injection du cône positif de H −1 dans W 1,q est compacte pour tout q < 2. J. Math. Pures Appl. 60 (1981), 309–22.Google Scholar
18Serre, D.. La compactité par compensation pour les systems hyperboliques non linéaires de deux equations à une dimension d'espace. J. Math. Pures Appl. 65 (1986), 423–68.Google Scholar
19Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics, Heriot–Watt Symposium, IV, ed. Knops, R. J., Pitman Research Notes in Mathematics, 136–92 (Harlow: Longman, 1979).Google Scholar
20Zhao, H. J. and Xuan, B. J.. Existence and convergence of solutions for the generalized BBM–Burgers equations with dissipatives term. Nonlinear Anal. 28 (1997), 1835–49.CrossRefGoogle Scholar
21Zhu, C. J.. Convergence of the viscosity solutions for 2 × 2 hyperbolic conservation laws with one characteristic field linearly degenerate on some zero measure sets. Chinese Sci. Bull. 40 (1995), 1639–43.Google Scholar
22Zhu, C. J.. Convergence of the viscosity solutions for the system of nonlinear elasticity (Preprint, 1994).Google Scholar